# Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems

## TL;DR

This paper explores the equivalence between determining finite Blaschke products from critical points, charge interaction models, and moment problems, revealing new theoretical insights and algorithmic possibilities.

## Contribution

It establishes the equivalence of three problems involving Blaschke products, charge configurations, and moment problems, and links these to polynomial solutions and energy minimization.

## Key findings

- Proves the equivalence of three key problems in complex analysis and mathematical physics.
- Connects Blaschke products with Stieltjes polynomials and moment problems.
- Suggests new algorithmic approaches based on energy minimization.

## Abstract

The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to other topics. Though existence and uniqueness of solutions are established for long, we present several new aspects which have not yet been explored to their full extent. In particular, we show that the following three problems are equivalent: (i) determining a finite Blaschke product from its critical points, (ii) finding the equilibrium position of moveable point charges interacting with a special configuration of fixed charges, (iii) solving a moment problem for the canonical representation of power moments on the real axis. These equivalences are not only of theoretical interest, but also open up new perspectives for the design of algorithms. For instance, the second problem is closely linked to the determination of certain Stieltjes and Van Vleck polynomials for a second order ODE and allows the description of solutions as global minimizers of an energy functional.